Abstract
This study investigates the horizontal transient response of partially embedded pile groups in multilayered transversely isotropic soils. The dynamic equation of pile groups is derived using the FEM by considering pile–pile interaction. The flexibility matrix is presented by applying the fundamental transient solutions of multilayered transversely isotropic soils. Then, the interaction solution between the piles and soils is obtained by using a FEM–BEM coupled method. The correctness of the present solution is validated by comparing the results with those in existing literature. Numerical examples are given to explore how free-standing length, pile–soil stiffness ratio, pile spacing ratio, soil’s transverse isotropy and stratification affect the horizontal transient response of partially embedded pile groups.
Abbreviations
- \(c\) :
-
Center distance between adjacent two piles
- \(d\), \(r\) :
-
Diameter and radius of the pile
- \(E_{\text{p}}\) :
-
Elastic modulus of the pile
- \(E_{\text{h}}\), \(E_{\text{v}}\) :
-
Elastic moduli of the soil in the horizontal plane and vertical direction, respectively
- \(G_{\text{v}}\) :
-
Shear modulus of the soil in the vertical direction
- \({\mathbf{f}}(t)\) :
-
Total load vector at the pile node with time
- \(f_{0}\) :
-
Intensity of the horizontal transient loading
- \(\{ f\}\), \(\{ f\}^{{\text{ext}}}\) :
-
Total load vector and external load vector acting at a pile node in the transformed domain, respectively
- \(\{ F\}^{m}\), \(\{ F\}^{{\text{ext}}}\) :
-
External load vectors acting at the \(m\)th pile and the pile group in the transformed domain, respectively
- \(h\) :
-
Thickness of the soil layer
- \(i\), \(j\) :
-
Numbering of the pile
- \(J_{1}\) :
-
Bessel function of the first order
- \([K]\), \([K]_{S}^{m}\), \([K]_{GS}\) :
-
Stiffness matrices for a pile element, the \(m\)th pile, and the pile group, respectively
- \([\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{K} ]_{GS}\), \([\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{K} } ]_{GS}\) :
-
Total stiffness matrices and extended stiffness matrix of a partially embedded pile group, respectively
- \(l\) :
-
Depth of the stratum
- \(L\) :
-
Length of the pile
- \({\text{L}}\) :
-
Symbol of Laplace integral
- \(L_{\text{S}}\) :
-
Length of the pile embedded in the soil
- \(L_{\text{F}}\) :
-
Length of the pile free-standing above the soil surface
- \([M]\), \([M]_{S}^{m}\), \([M]_{GS}\) :
-
Mass matrices for a pile element, the \(m\)th pile and the pile group, respectively
- \(M_{h}\) :
-
Dimensionless moment
- \(n_{F}\), \(n_{S}\) :
-
Number of pile nodes above the soil surface and embedded in the soil, respectively
- \(N\) :
-
Number of piles in a pile group
- \(\rho_{\text{p}}\), \(\rho_{\text{s}}\) :
-
Densities of the pile and the soil, respectively
- \(\{ q\}\), \(\{ q\}_{S}^{m}\), \(\{ q\}^{p}\) :
-
Reaction load vectors acting at a pile node, the \(m\)th pile and the pile group, respectively
- \(\{ q\}_{GS}\) :
-
Load vector acting in the soil
- \([Q]\), \([Q]_{S}^{m}\), \([Q]_{GS}\) :
-
Load transfer matrices for a pile node, the \(m\)th pile and the pile group, respectively
- \([R]_{GF}\), \([R]_{S}^{i}\), \([R]_{GS}\) :
-
Soil flexibility matrices for a fully embedded pile group, a partially embedded single pile and the partially embedded pile group, respectively
- \(s\) :
-
Laplace parameter
- \(t\) :
-
Real time
- \([T]_{GS}\), \([\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{T} ]_{GS}\) :
-
Combined matrix and extended combined matrix with load transfer matrix and soil flexibility matrix
- \({\mathbf{u}}(t)\) :
-
Displacement and rotation vector at the pile node with time
- \({\ddot{\mathbf{u}}}(t)\) :
-
Second derivative of \({\mathbf{u}}(t)\) with time
- \(\{ u\}\), \(\{ U\}^{m}\), \(\{ U\}_{GF}\), \(\{ U\}_{GS}\) :
-
Displacement and rotation vectors in the transformed domain for a pile node, the \(m\)th pile, a fully embedded pile group and a partially embedded pile group, respectively
- \(u_{r}\), \(u_{\theta }\), \(U_{ij}\) :
-
Displacement components and the total displacement of the \(j\)th pile in cylindrical coordinates, respectively
- \(U_{0}\), \(U_{2}\) :
-
Coupled displacement components of the \(j\)th pile in the Laplace–Hankel domain
- \(U_{h}\) :
-
Dimensionless horizontal displacement
- \(\mu_{\text{h}}\), \(\mu_{\text{vh}}\) :
-
Poisson’s ratios of the soil in the horizontal and vertical plane, respectively
- \(V_{h}\) :
-
Dimensionless shear force
- \(Z\) :
-
Depth of the pile
- \(\tau\) :
-
Dimensionless time
- \(\xi\) :
-
Hankel integral parameter
- \(\{ \theta \}^{m}\) :
-
Rotation of the \(m\)th pile in a partially embedded pile group
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Acknowledgements
This research is supported by the National Natural Science Foundation of China (Grant No. 41672275). The authors would like to thank the editor Prof. Jian Chu and the reviewers for their valuable comments that contribute to improving the quality of this paper.
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Li, Y., Ai, Z.Y. Horizontal transient response of a pile group partially embedded in multilayered transversely isotropic soils. Acta Geotech. 16, 335–346 (2021). https://doi.org/10.1007/s11440-020-01023-6
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DOI: https://doi.org/10.1007/s11440-020-01023-6